• Here we propose a new method for classifying aerial stems of woody plants by developmental stage, using the logarithmic reciprocal of relative growth rate (LRR) as an indicator of developmental stage.
• Stem analyses were conducted on naturally dead aerial stems of Lindera umbellata to clarify the changes in LRR over a lifetime. LRR, number of current-year shoots, and the recruitment and mortality rates of shoots of living stems were investigated.
• LRR was at a minimum value at age 1 yr and at a maximum just before each stem died. There was little difference between the ranges of stem LRR. The recruitment and mortality rates of shoots depended on LRR.
• LRR satisfied the necessary and sufficient condition for a variable as an indicator of stage better than either age or size. The LRR-structured model accurately demonstrates the real demographic processes in shoot populations over a lifetime of aerial stems. This result supports the utility of LRR as an indicator of stage. The method using LRR can be applied to analyses for other growth processes.
Ishii & Takeda (1997) described changes in the number of current-year shoots with stem age for the clonal shrub Hydrangea hirta, estimating shoot demographic parameters from the stem population classified by stem-age class in the field. In this age-structured model it is assumed that demographic parameters of the shoot population depend on stem age, and that there is little difference between these parameters in stems in the same age class. However, this assumption is not appropriate, as an age class of H. hirta contained shrubs in both mature and senescence stages, because of variations in stem longevity. Thus, although Ishii & Takeda (1997) could describe shoot demography during the period from sapling to mature stages, in which the number of current-year shoots peaked, they could not describe the senescence stage, in which the number of shoots decreased approaching death.
Also, size is not always adequate as an indicator of developmental stage for woody plants. As these plants undergo little change in size for a long period after reaching the mature stage, it is difficult to arrange individuals from mature to senescence stages using size as an indicator of stage. Moreover, the maximum sizes of individuals differ depending on the environment. For these reasons, a size class in the size-structured model contains individuals of both mature and senescence stages.
Here we assume the necessary and sufficient conditions for a variable as an indicator of stage are that: (1) the variable takes a characteristic value just after an individual emerges and takes another characteristic value just before it dies; and (2) there is no difference in the range of the variable among different individuals. For a population that consists of individuals with different maximum sizes and longevities, age or size satisfy the first condition, but not the second. Therefore these variables are not sufficient as indicators of stage.
We propose a new method for classifying aerial stems of woody plants by developmental stage using relative growth rate (RGR) as an indicator of stage. The biological rationale for this approach is as follows. For plants, the growth of the shoots and roots is a result of the formation of new cells by embryonic tissues of the apices, called apical meristems. The shoot apical meristem, which consists of a small number of dividing cells, gives rise to all of the other cells and tissues in the primary shoot. In aerial stems of woody plants, secondary tissues arise by the activity of lateral meristems (vascular cambium) produced by the apical meristem. The vascular cambium adds to the stem diameter by adding new xylem cells to the inside and new phloem cells to the outside. Based on these characteristics of growth, the proportion of the number of cells in meristems to that in the stem is the largest just after the stem originated, over a lifetime of the stem. However, although a stem grows in thickness with secondary growth, the vascular cambium does not grow in thickness because it is always retained as a thin layer of cells beneath the surface of the stem. Therefore the growth rate in volume of the vascular cambium is slower than that of the stem. As a result, the proportion of the number of cells in meristems to that in the stem decreases with increasing stem size. Although meristem cells actively divide and, as a result, new cells are continually added to the stem body at the young stages, their activities decline with the progress of senescence, and eventually the stem dies. For these reasons, the RGR of a stem decreases with its increasing size and with the progress of senescence. If we assume that an aerial stem originates from a shoot apical meristem of infinitesimal size, the instantaneous RGR reaches infinity just after the stem originates. On the other hand, instantaneous RGR should reach infinitesimal just before the stem dies, because the rate of cell divisions in the meristems also becomes infinitesimal. Based on this assumption, the RGR of stems satisfies both conditions, and stems can therefore be classified by developmental stages using RGR.
We used the clonal shrub Lindera umbellata to examine whether RGR satisfies the two conditions described. Although it is ideal that RGR is calculated by an instantaneous measure, it is difficult to use for methodological reasons. As the growth rate of woody plant stems varies seasonally, the stage of a stem cannot be represented by instantaneous RGR in one season. In the present study, therefore, RGR was calculated by measuring in one growing season.
If RGR satisfies the two conditions described, it must be a better indicator of stage than either age or size, and various growth processes over the lifetime of aerial stems would be analysed better by using RGR than by age or size. The aims of this study were (1) to show that the RGR values for aerial stems of L. umbellata are at a maximum just after they have emerged and at a minimum just before they die; and (2) to determine whether there is only a small difference in ranges of RGR among stems with different maximum sizes and longevities. A further aim was (3) to show that this method of classifying aerial stems by stage using RGR is useful for analyses of stem growth processes. We examined whether the demographic processes of shoot populations of aerial stems are determined depending on stem RGR.
Materials and Methods
Lindera umbellata Thunb. (Lauraceae) is a dioecious, broad-leaved, deciduous shrub that is commonly distributed on the forest floor of cool-temperate forests in Japan (Sasaki, 1970). A single shrub consists of many aerial stems of various ages and sizes, growing in a tight clump from a single underground root collar. The clumps maintain themselves for a long time by replacing old stems with new ones that occasionally sprout from the root collar. Maturation from seedlings to clumps with multiple stems takes > 10 yr (Yanagisawa & Kawanishi, 1951). The maximum size of aerial stems is c. 6 m high and 7 cm in basal diameter, and the maximum age is c. 30 yr (Tamai & Tempo, 1990). The stem usually has many branches, and branches consist of many shoots.
The study was carried out in a secondary forest dominated by Quercus serrata Thunb. ex Murray. at Hiruzen Experimental Forest Station of Tottori University (35°17′ N, 133°34′ E), in the northern part of Okayama Prefecture, Chugoku District, Japan. At this location L. umbellata is a common understorey shrub. Meteorological records obtained from a station c. 3 km from this location showed that the mean annual temperature and mean annual precipitation are 11.3°C and 2140 mm, respectively.
The study was carried out on 75 living and 19 dead aerial stems in 21 clumps of L. umbellata. In mid-April (the beginning of the growing season) 2002, the stem length (m) and two diameters (cm) in perpendicular angles at intervals of 20 cm from the base toward the top were measured in each of the 75 living stems. In early October 2002 (the end of the growing season) these measurements were repeated and the number of current-year shoots in each stem was counted. Dead aerial stems were analysed to determine changes over a lifetime in the RGR of aerial stems of L. umbellata. The 19 dead aerial stems were all collected from the study site. The stem length (m) and basal diameter (cm) of each dead stem were measured, then cross-sectional disks of each stem at intervals of 20 cm from base to top were obtained for stem analysis. The widths of wood rings along each of four radii of two diameters perpendicular with each other in each disk were measured using a microscope. The arithmetic mean of the four radii was used for calculation of stem volume. Smalian's formula for the portions of stems shaped like a frustum of a paraboloid and for a circular cone for the tips were used to estimate stem volume in each age class (Nagumo & Minowa, 1990). The stem volumes of living stems were also estimated by the same formulae using size data obtained at the start and end of the growing season.
To analyse the demographic processes of shoot populations, it is necessary to investigate the recruitment and mortality rates of shoots per stem. This investigation was carried out for 19 of the 75 living stems with different RGRs. We defined a shoot as each terminal, unbranched segment of a branch, which is equivalent to the first-order branch in centripetal ordering systems such as the Strahlar system (Borchert & Slade, 1981). The shoots were classified into three types: newly recruited shoots; newly dead shoots; and other living shoots. Newly recruited shoots were defined as sylleptic shoots that had emerged on current-year shoots; newly dead shoots as shoots that had died in the current year. In L. umbellata, the number of newly dead shoots can be determined by counting the number of bud-scale scars. As bud breaks occur once a year and there are few shoots initiated from dormant buds in the stems, the numbers of bud scale scars from branch base to terminal buds of newly dead shoots and neighbouring living shoots are not different in most cases. Therefore we determined the number of newly dead shoots by comparing the number of bud scale scars from branch base to each dead shoot, with the number of bud scale scars from branch base to neighbouring living shoots. The number of each shoot type in the 19 stems was counted in early October 2002 (end of the growing season).
where V(t + 1) (cm3) and V(t) (cm3) are stem volume in year t + 1 and stem volume in year t, respectively. Here we define LRR (the logarithmic reciprocal RGR) as:
LRR = ln((RGR + a)−1)(Eqn 2)
where a is a fixed value to prevent the value of LRR from being infinity when RGR = 0. In the present study, this was determined 0.001 as not effect the minimum value in LRR of each stem.
Variations in size, age and LRR among individuals were examined. Each stage of each individual was represented by the relative position in the range of each variable (LRR, age and stem volume). We defined the relative position as 0 when each variable of each individual has a minimum value, and as 1 when the variable reaches a maximum value. From stem analysis data we obtained values for each variable and relative positions for each year over the lifetimes of the 19 dead stems. The relationships between each variable and the relative position for dead stems were regressed by the following equation:
Y = b + cX(Eqn 3)
where b and c are regression constants, X is LRR, age or log [stem volume (cm3)], and Y is the relative position. The constants were obtained by applying a linear regression program that minimizes the sum of squares of residuals (RSS) in terms of Y.
The recruitment rate of shoots on each stem was estimated as the percentage in the number of newly emerged shoots in the current year to that of living shoots in the previous year. The mortality rate of shoots on each stem was estimated as the percentage in the number of newly dead shoots in the current year to that of living shoots in the previous year. The relationships between LRR and recruitment rate, and between LRR and mortality rate of shoots, were regressed by the following equations:
r = d exp(fLRR)(Eqn 4)
m = d′ exp(f′LRR)(Eqn 5)
where d, d′, f and f′ are regression constants, and r and m are recruitment rate (% year-1) and mortality rate (% year-1) of shoots, respectively. The constants were obtained by applying a linear regression program that minimizes the sum of squares of residuals (RSS) in terms of ln r and ln m. The growth rate of a shoot population was estimated by subtracting the mortality rate from the recruitment rate of shoots. Similarly, the expectation curve was obtained by subtracting the regression equation of the mortality rate from that of the recruitment rate of shoots.
g = defLRR − d′ef′LRR(Eqn 6)
where g is the growth rate (% year-1) of a shoot population. Using equation (6) and LRR measured by stem analysis, we simulated the changes over a lifetime in the number of current-year shoots on the 19 dead stems. The model equation is:
N(t + 1) = N(t)g(t)(Eqn 7)
where N(t) is the number of current-year shoots in year t and g(t) is the growth rate of the shoot population in year t. Although each stem actually has one current-year shoot at age 0, it usually grows with no branching during the first several years. In the present study, however, we could not predict the growth rates of shoot populations during this period because the data for stems in these age classes are not included in the data to predict the growth rates of shoot populations by LRR. Therefore, in the simulation model, it was assumed that the initial value of the number of current-year shoots, N(0) (i.e. number of current-year shoots at age 0) is 0.01. Based on this assumption, the number of current-year shoots increased beyond 1 at age 3.4 ± 0.7 yr (mean ± SD) in each stem.
The 19 sampled aerial stems that died naturally in the field varied in longevity and maximum size: longevity is 16.8 ± 7.5 yr (mean ± SD), and stem length and stem volume are 3.54 ± 1.36 m and 114.3 ± 139.6 cm3, respectively. The LRRs of all stems were minimum at the age of 1 yr (−0.16 ± 0.10 SD) and then increased with advancing age (Fig. 1), finally reaching maximum values just before the plants died (2.12 ± 0.35 SD).
The relationships of LRR, age and stem volume to relative position were significant (ancova, P < 0.001, Fig. 2). However, the coefficient of determination was higher in the relationship between LRR and relative position (r2 = 0.91) than in the relationships between age and relative position (r2 = 0.71); and log (stem volume) and relative position (r2 = 0.58).
To predict LRR-dependent changes in demographic parameters of shoot populations, we determined the relationships of stem LRR with the recruitment rate and mortality rate of shoots (Fig. 3a,b, respectively). The recruitment rate of shoots decreased significantly with increase in LRR (t-test, r2 = 0.86, P < 0.001; Fig. 3a). Recruitment rate decreased rapidly from 733 to 18.5% for LRR ≤ 1.0, and recruitment rates were all nearly 0 for LRR > 1.0 (0.0–10.8% yr-1). On the other hand, the mortality rate of shoots increased with increasing LRR (r2 = 0.86, P < 0.001; Fig. 3b). Mortality rates were negligible for LRR ≤ 0.6 (0.0–2.1% yr-1), then increased with increasing LRR from 0.6. The mortality rate was highest (70.6% yr−1) when LRR was > 2.0. Consequently, the growth rate of shoot populations decreased significantly with increasing LRR (r = −0.83, P < 0.01; Fig. 3c). The rates decreased rapidly from 733 to 15.9% for LRR ≤ 1.0; were 3.4–13.4% when LRR ranged from 1.0 to 1.5; then became negative for LRR > 1.5 (from 5.1 to −70.6% yr−1).
The changes over a lifetime in the number of current-year shoots per stem were simulated for the 19 dead stems using the growth rates of shoot populations derived from the expectation curve shown in Fig. 3c. The number of current-year shoots per stem increased toward stable values with increasing LRR (Fig. 4). The number of current-year shoots on stems was maximum at an LRR of 1.29 ± 0.14 (mean ± SD), then began to decrease with increasing LRR.
The relationship between LRR and number of current-year shoots for the 75 living stems is shown in Fig. 5. The number of current-year shoots per stem increased with increasing LRR when LRR was < 1. The number of current-year shoots was optimal at an LRR of c. 1.3. At LRR > 1.5, the number of current-year shoots tended to decrease with increasing LRR. This pattern of scatter distribution is compatible with the projected growth curves in the number of current-year shoots with increasing LRR in the dead stems (Fig. 4).
We first assumed the necessary and sufficient condition for a variable as an indicator of developmental stage. The first condition, that of the initial and final values being extreme values, is a necessary and sufficient condition for a variable as an indicator of developmental stage at the individual stem level. If this condition is satisfied by an individual stem, the developmental stage of the stem can be represented by the relative position between the two extreme values. As shown in Fig. 1, it is evident that LRR, age and stem volume satisfy this condition. Therefore LRR, age and stem volume are useful as indicators of developmental stage at the individual stem level.
However, if a variable is used as the common indicator of stage among different stems, the ranges of variables in different stems must be the same, at least within a population of stems. This is the second condition: the range of variables must be common for different stems. If this requirement is satisfied, a certain relative position between minimum and maximum values can be represented by the same value on the same variable axis for different stems. As a result, these stems can be arranged by the variable on the successive course of stage over lifetime. In the present study, we showed what proportions of the variation in relative position are explained by the variations in LRR, age and log(stem volume). LRR predicted the relative position for various stems most precisely, while variations in age or log(stem volume) were great. The coefficient of determination was highest in the relationship between LRR and relative position (Fig. 2). Therefore we conclude that, in aerial stems of L. umbellata, LRR is a better indicator of stage than age or size, because LRR satisfies the second condition better than either age or size.
To determine whether the LRR-structured model is useful for describing the demographic processes in shoot populations over a lifetime of aerial stems of L. umbellata, we simulated the changes over a lifetime in the number of current-year shoots for the 19 dead stems, based on the changes over a lifetime in LRR (Fig. 4). The results of this simulation accurately demonstrated the real demographic processes in shoot populations of L. umbellata. It is reasonable that the number of current-year shoots per stem increases as saplings grow, then decreases in the senescence stage after they have reached the peak of development in the mature stage. Moreover, the pattern of scatter distribution in the relationship between number of current-year shoots per living stem and its LRR (Fig. 5) is compatible with the projected growth curves in the number of current-year shoots for the dead stems.
LRR would be a better indicator of stage, for analysis of demographic processes of shoot populations on aerial stems of woody plants, than age or size. In the age-structured model, Ishii & Takeda (1997) could describe demographic processes of shoot populations from sapling to mature stage when the crown expansion stopped, for aerial stems of Hydrangea hirta. However, they failed to describe the demographic processes of shoot populations from the mature stage to the senescence stage. In their age-structured model, an age class contained stems of both mature and senescence stages, because the longevity of stems varied. Similarly, in woody plants stem size is unlikely to be adequate as an indicator of stage for predicting the demographic parameters of shoot populations. As there is little change in the sizes of woody plants after the stems have reached the mature stage, it is difficult to arrange stems from mature to senescence stages using stem size as an indicator of stage. Moreover, maximum sizes of stems differ depending on the environment. Thus a size class in the size-structured model contains stems of both mature and senescence stages. On the other hand, the present study clarifies that the LRR-structured model can describe the demographic processes of shoot populations not only from sapling stage to mature stage, but also from mature stage to senescence stage, even if aerial stems of L. umbellata are variable in terms of longevity and maximum size. Therefore we conclude that the LRR-structured model is more useful than a model structured by age or size for predicting shoot demographic parameters for aerial stems of woody plants.
We wish to thank the staff of Hiruzen Forest Station for granting us permission to work in Hiruzen Forest. We also wish to thank the members of the Laboratory of Forest Biology, Faculty of Agriculture, Kyoto University for their comments and suggestions. Funding for D. Fujiki was provided by the Japan Scholarship Society.